Why is part of object's shadow sliced?
This is because the shadows are only projected from geometry within a limited area around the sun lamp. To visualize this area, enable Show Shadow Box in the lamp's settings:



This makes the cause much clearer:



The solution is to make the sun lamp follow your character (or camera) so that objects near the view are always within the "frustum". 

The simplest way to do this is by vertex parenting the lamp to your model:


Select the lamp
Select the model
Switch to edit mode (&#8633; Tab)
Select a single vertex
Press &#9096; CtrlP and select Vertex Parent




Now the lamp will follow that vertex's location without rotating:



Projecting shadows is relatively memory and processing intensive, so the smaller this area is, the better. You can adjust the size with the clipping settings:



You'll likely want to optimize the frustum size for what is visible from whatever camera view you end up with, such that the shadow boundary isn't in the foreground and overtly apparent.